A002961 -id:A002961 - cp's OEIS frontend

A231546 Numbers k such that sigma(k) = sigma(k-1).

15, 207, 958, 1335, 1365, 1635, 2686, 2975, 4365, 14842, 18874, 19359, 20146, 24958, 33999, 36567, 42819, 56565, 64666, 74919, 79827, 79834, 84135, 92686, 109215, 111507, 116938, 122074, 138238, 147455, 161002, 162603, 166935, 174718, 190774, 193894, 201598

Offset: 1

Jaroslav Krizek, Nov 11 2013

nonn

Also, numbers k such that k = antisigma(k) - antisigma(k-1), where antisigma(k) = A024816(k) = the sum of the non-divisors of k that are between 1 and k.

			15 = antisigma(15) - antisigma(14) = 96 - 81.

Amiram Eldar, Table of n, a(n) for n = 1..10135 (calculated using the b-file at A002961; terms 1..550 from Hugo Pfoertner)

Cf. A002961, A024816 (antisigma(n)), A231545 (numbers n such that antisigma(n) = antisigma(n-1)).

SequencePosition[DivisorSigma[1,Range[210000]],{x_,x_}][[;;,2]] (* Harvey P. Dale, May 28 2024 *)

n=0;sp=sigma(2);for(k=3,oo,my(s=sigma(k));if(s==sp,print1(k,", ");n++;if(n>36,break));sp=s) \\ Hugo Pfoertner, Mar 06 2020

a(n) = A002961(n) + 1.

More terms from Hugo Pfoertner, Mar 06 2020

A238380 Numbers k such that the average of the divisors of k and k+1 is the same.

Original entry on oeis.org

5, 14, 91, 1334, 1634, 2685, 3478, 5452, 9063, 13915, 16225, 20118, 20712, 33998, 42818, 47795, 64665, 79338, 84134, 103410, 106144, 109214, 111683, 122073, 123497, 133767, 166934, 170884, 203898, 224561, 228377, 267630, 289454, 383594, 384857, 391348, 440013

Offset: 1

Giovanni Resta, Feb 25 2014

nonn

The average of the divisors of n is equal to sigma(n)/tau(n).

Up to 5*10^12, there are only 3 terms for which the mean is not an integer, namely 254641594575, 280895287491 and 328966666100.

			91 is a term since the average of the divisors of 91 and 92 is the same. Indeed, (1+7+13+91)/4 = (1+2+4+23+46+92)/6.

Giovanni Resta, Table of n, a(n) for n = 1..6934 (terms < 5*10^12)

Cf. A002961.

av[n_] := DivisorSigma[1,n]/DivisorSigma[0,n]; Select[Range[10^5], av[#] == av[# + 1] &]
SequencePosition[Table[DivisorSigma[1,n]/DivisorSigma[0,n],{n,450000}],{x_,x_}][[All,1]] (* Harvey P. Dale, Jun 01 2022 *)

from sympy import divisors
from fractions import Fraction
def aupto(limit):
  alst, prev_divavg = [], 1
  for n in range(2, limit+2):
    divs = divisors(n)
    divavg = Fraction(sum(divs), len(divs))
    if divavg == prev_divavg: alst.append(n-1)
    prev_divavg = divavg
  return alst
print(aupto(440013)) # Michael S. Branicky, May 14 2021

A058072 Numbers k such that sigma(k) divides sigma(k+1), where sigma(k) is sum of positive divisors of k.

Original entry on oeis.org

1, 5, 14, 125, 206, 957, 1253, 1334, 1364, 1634, 1673, 1919, 2685, 2759, 2974, 3127, 4364, 5191, 7615, 11219, 12035, 14841, 18873, 19358, 20145, 24957, 27089, 33998, 36566, 42818, 43817, 47795, 48559, 49955, 50039, 56564, 56975, 58373, 58463

Offset: 1

Leroy Quet, Nov 11 2000

nonn

The quotient (sigma(k+1)/sigma(k)) is equal to 1, 2, 3, 4 or 5 for the first 5000 terms. - Donovan Johnson, Oct 21 2012

			5 is included because sigma(5) = 6 divides sigma(6) = 12.

Donovan Johnson, Table of n, a(n) for n = 1..5000

Cf. A000203, A002961.

Select[Range[10^5], Divisible[DivisorSigma[1, # + 1], DivisorSigma[1, #]] &] (* Michael De Vlieger, Sep 03 2017 *)

isok(n) = !(sigma(n+1) % sigma(n)); \\ Michel Marcus, Sep 04 2017

More terms from Benoit Cloitre, Jul 27 2002

A324295 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

Original entry on oeis.org

2, 3, 4, 186, 318, 434, 473, 582, 730, 978, 1024, 1035, 1245, 1357, 1397, 1506, 1661, 1902, 2085, 2116, 2224, 2329, 2453, 2505, 2506, 2770, 2954, 3144, 3345, 3377, 3624, 3641, 3765, 3790, 3882, 4037, 4172, 4438, 4898, 4938, 4975, 5221, 6126, 6285, 6312, 6356

Offset: 1

Amiram Eldar, Sep 03 2019

nonn

			186 is in the sequence since A070039(186) = A070039(187) = 12.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002961, A064115, A064125, A070039, A164522, A164557, A293183, A306985, A325175.

s[n_] := DivisorSum[n, # &, # < Sqrt[n] &]; seq={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 6500}]; seq

A338452 Numbers k such that k and k+1 have the same total binary weight of their divisors (A093653).

Original entry on oeis.org

3, 4, 7, 20, 31, 57, 94, 98, 118, 122, 127, 201, 213, 218, 230, 242, 243, 244, 334, 384, 393, 423, 429, 481, 565, 603, 633, 694, 704, 729, 766, 844, 921, 1138, 1141, 1221, 1262, 1401, 1533, 1654, 1726, 1761, 1837, 1838, 1862, 1882, 1942, 2162, 2245, 2361, 2362

Offset: 1

Amiram Eldar, Oct 28 2020

Numbers k such that A093653(k) = A093653(k+1).

The Mersenne primes (A000668) are terms since if 2^p - 1 is a prime then A093653(2^p-1) = A093653(2^p) = p+1.

			3 is a term since A093653(3) = A093653(4) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000120, A093653.
A000668 is a subsequence.
Similar sequences: A002961, A005237, A054004, A064125, A238380, A293183, A306985.

f[n_] := DivisorSum[n, DigitCount[#, 2, 1] &]; s = {}; f1 = f[1]; Do[f2 = f[n]; If[f1 == f2, AppendTo[s, n - 1]]; f1 = f2, {n, 2, 240}]; s

A053215 Sum of divisors of those numbers n such that n and n+1 have the same sum of divisors.

Original entry on oeis.org

24, 312, 1440, 2160, 2688, 2640, 4320, 4464, 7644, 22932, 28314, 29040, 34560, 37440, 51840, 56160, 65280, 100800, 115200, 114912, 120960, 120960, 138240, 153216, 194400, 168960, 178560, 186048, 207360, 221184, 244800, 280800, 276480

Offset: 1

Asher Auel, Jan 11 2000

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10135 (from the b-file at A002961; terms 1..4804 from T. D. Noe)

Cf. A000203, A002961, A053249.

Transpose[Select[{DivisorSigma[1,First[#]],DivisorSigma[1,Last[#]]}&/@ Partition[Range[280000],2,1],First[#]==Last[#]&]][[1]] (* Harvey P. Dale, May 07 2012 *)
DivisorSigma[1,#]&/@SequencePosition[DivisorSigma[1,Range[280000]],{x_,x_}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 21 2017 *)

a(n) = sigma(A002961(n))

A077087 Numbers k such that sigma(k+1) = 3 * sigma(k).

Original entry on oeis.org

1, 1919, 2759, 11219, 27089, 50039, 58463, 100127, 113831, 115289, 120203, 131879, 148511, 233729, 244319, 308039, 461099, 554063, 596447, 1406303, 1486619, 2285519, 2880989, 5138783, 5369111, 5521619, 5736743, 6621383, 7496279, 7683191, 8571527, 8848619

Offset: 1

Labos Elemer, Oct 31 2002

nonn

			k = 1: sigma(2)/sigma(1) = 3/1 = 3;
k = 9563231: sigma(k+1)/sigma(k) = 31026240/10342080 = 3.

Donovan Johnson, Table of n, a(n) for n = 1..500

Cf. A000203, A002961, A067081, A077086, A272027 (3*sigma(n)).

Do[s=Mod[a=DivisorSigma[1, n+1], b=DivisorSigma[1, n]]; If[Equal[s, 0]&&Equal[a/b, 3], Print[n]], {n, 1, 10000000}]
Position[Partition[DivisorSigma[1,Range[9*10^6]],2,1],?(3#[[1]] == #[[2]]&),1,Heads->False]//Flatten (* _Harvey P. Dale, Oct 03 2017 *)

A163193 Numbers k such that sigma(k) = 2*sigma(k+1).

Original entry on oeis.org

12, 70, 88, 204, 220, 1750, 1888, 2958, 8142, 8632, 9114, 14664, 18414, 18762, 20118, 20712, 25194, 45520, 64206, 65652, 65964, 77814, 79338, 79824, 85096, 90804, 103410, 103644, 117822, 158946, 163938, 176364, 185776, 186612, 194416, 202656, 203898, 245632

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 22 2009

nonn

The cases sigma(k) = 3*sigma(k+1) are rarer: k=180, 12000, 30996, 47940, ... [R. J. Mathar, Jul 25 2009]

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A067081, A002961, A058073.

f[n_]:=DivisorSigma[1,n]; lst={};Do[If[f[n]==f[n+1]*2,AppendTo[lst,n]], {n,9!}];lst

{k: A000203(k) = 2*A000203(k+1)}.

Edited by R. J. Mathar, Jul 25 2009

A164522 Numbers k such that sigma_odd(k) = sigma_odd(k+1), where sigma_odd(k) is the sum of the odd divisors of k (A000593).

Original entry on oeis.org

1, 27089, 115289, 233729, 2529090, 2880989, 14059709, 17192909, 17540250, 18693990, 34902630, 54722249, 58517910, 82200689, 83087730, 92991990, 93623250, 93862230, 96578369, 111681990, 112244369, 155120129, 206450369, 269626769, 293182469, 303206310, 324764910

Offset: 1

Amiram Eldar, Aug 12 2019

nonn

			27089 is in the sequence since A000593(27089) = A000593(27089 + 1) = 27456.

Amiram Eldar, Table of n, a(n) for n = 1..100
Daeyeoul Kim, Nazli Yildiz Ikikardes, Yan Li, and Lianrong Ma, On the Problem sigma_od(n) = sigma_od(n+ 1), Filomat, Vol. 33, No. 2 (2019), pp. 543-559.

Cf. A000593, A002961, A064115, A064125, A293183, A306985.

v:=[&+[d:d in Divisors(m)|IsOdd(d)] :m in [1..5000000]]; [k:k in [1..#v-1]| v[k] eq v[k+1]]; // Marius A. Burtea, Aug 12 2019

f[p_, e_] := If[p == 2, 1, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); s1=0; seq={}; Do[s2 = s[n]; If[s2 == s1, AppendTo[ seq, n-1]]; s1 = s2, {n, 1, 10^6}]; seq

A164557 Numbers k such that s(k) = s(k+1), where s(k) is the sum of divisors d of k such that k/d is odd (A002131).

Original entry on oeis.org

3, 6, 7, 10, 22, 31, 46, 58, 69, 82, 106, 127, 140, 154, 160, 166, 178, 226, 262, 286, 346, 358, 382, 466, 478, 502, 562, 586, 718, 748, 781, 838, 862, 886, 982, 1001, 1018, 1066, 1186, 1282, 1299, 1306, 1318, 1366, 1438, 1486, 1522, 1614, 1618, 1672, 1704, 1822

Offset: 1

Amiram Eldar, Aug 12 2019

nonn

			3 is in the sequence since A002131(3) = A002131(3 + 1) = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Daeyeoul Kim and Abdelmejid Bayad, Convolution identities for twisted Eisenstein series and twisted divisor functions, Fixed Point Theory and Applications 2013, No. 1 (2013), Article 81, alternative link.
Daeyeoul Kim, Nazli Yildiz Ikikardes, Yan Li, and Lianrong Ma, On the Problem sigma_od(n) = sigma_od(n+ 1), Filomat, Vol. 33, No. 2 (2019), pp. 543-559.

Cf. A002131, A002961, A064115, A064125, A293183, A306985.

v:=[&+[d:d in Divisors(m)|IsOdd(Floor(m/d))] :m in [1..2000]]; [k:k in [1..#v-1]| v[k] eq v[k+1]]; // Marius A. Burtea, Aug 12 2019

f[p_, e_] := If[p == 2, p^e, (p^(e+1)-1)/(p-1)]; s[1] = 1; s[1] = 1; s[n_] := Times @@ (f @@@ FactorInteger[n]); s1=0; seq={}; Do[s2 = s[n]; If[s2 == s1, AppendTo[seq, n-1]]; s1 = s2, {n, 1, 2000}]; seq

cp's OEIS Frontend

A231546 Numbers k such that sigma(k) = sigma(k-1).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A238380 Numbers k such that the average of the divisors of k and k+1 is the same.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A058072 Numbers k such that sigma(k) divides sigma(k+1), where sigma(k) is sum of positive divisors of k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A324295 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A338452 Numbers k such that k and k+1 have the same total binary weight of their divisors (A093653).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A053215 Sum of divisors of those numbers n such that n and n+1 have the same sum of divisors.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A077087 Numbers k such that sigma(k+1) = 3 * sigma(k).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

A163193 Numbers k such that sigma(k) = 2*sigma(k+1).

Views

Author

Keywords

Comments

Links